A sequence of works [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} provided lower bounds for diameter approximation, based on the Strong Exponential Time Hypothesis (SETH) [8]}, [9]} that CNF-SAT on \(n\)  variables and \(O(n)\)  clauses requires \(2^{n-o(n)}\)  time. The first such lower bound by [1]} showed that any \(3/2-\varepsilon \)  approximation to the diameter of a directed or undirected unweighted graph for \(\varepsilon >0\) , running in \(O(m^{2-\delta })\)  time for \(\delta >0\) , would refute SETH, and hence the [1]} \(3/2\) -approximation algorithm has a (conditionally) optimal approximation ratio for a subquadratic time algorithm for diameter.
Later, Backurs, Roditty, Segal, Vassilevska W. and Wein [2]} showed that under SETH, any \(O(m^{3/2-\varepsilon })\)  time algorithm can at best achieve a \(1.6\) -approximation to the diameter of an undirected unweighted graph. Thus, the [1]} \(3/2\) -approximation algorithm has a (conditionally) optimal approximation running time for a \(1.6-\delta \) -approximation algorithm.
